Find the volume of the parallelepiped constructed on vec…

Find the volume of the parallelepiped constructed on $\vec a=\{0,1,1\}$, $\vec b=\{1,0,1\}$, $\vec c=\{1,1,0\}$.

Explanation of Method

The volume of a parallelepiped.

\[ [\vec a,\vec b,\vec c]=\begin{vmatrix}0&1&1\\1&0&1\\1&1&0\end{vmatrix}=0-1(0-1)+1(1-0)=2. \]

Check: $\vec a\times\vec b=\{1,1,-1\}$ and $\{1,1,-1\}\cdot\vec c=2$ $\checkmark$

\[V=|[\vec a,\vec b,\vec c]|=2\]

The triple is right-handed (positive sign). The tetrahedron on the same three vectors has volume $\tfrac{2}{6}=\tfrac13$.

$O$ $\vec a$ $\vec b$ $\vec c$ $(2,2,2)$

The parallelepiped on $\vec a,\vec b,\vec c$ (the face diagonals of the unit cube from $O$); its volume is $2$.

\[\boxed{\,V=|[\vec a,\vec b,\vec c]|=2\,}\]